Continuum-extrapolated NNLO valence PDF of the pion at the physical point

Abstract

We present lattice QCD calculations of the valence parton distribution function (PDF) of pion employing next-to-next-leading-order (NNLO) perturbative QCD matching. Our calculations are based on three gauge ensembles of $2+1$ flavor highly improved staggered quarks and Wilson-Clover valance quarks, corresponding to pion mass ${m}_{\ensuremath{\pi}}=140\text{ }\text{ }\mathrm{MeV}$ at a lattice spacing $a=0.076\text{ }\text{ }\mathrm{fm}$ and ${m}_{\ensuremath{\pi}}=300\text{ }\text{ }\mathrm{MeV}$ at $a=0.04$, 0.06 fm. This enables us to present, for the first time, continuum-extrapolated lattice QCD results for the NNLO valence PDF of the pion at the physical point. Applying leading-twist expansion for renormalization group invariant (RGI) ratios of bilocal pion matrix elements with NNLO Wilson coefficients, we extract second, fourth, and sixth Mellin moments of the PDF. We reconstruct the Bjorken-$x$ dependence of the NNLO PDF from real-space RGI ratios using a deep neural network as well as from momentum-space matrix elements renormalized using a hybrid scheme. All our results are in broad agreement with the results of global fits to the experimental data carried out by the xFitter and JAM collaborations.